Lastly, Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things.[782] Let us consider these statements in detail. Footnote 780: Arist. _Met._ Α, 5. 987 a 15. Footnote 781: _Met. ib._ 986 a 15 (R. P. 66). Footnote 782: _Met._ Α, 6. 987 b 27, ὁ μὲν (Πλάτων) τοὺς ἀριθμοὺς παρὰ τὰ αἰσθητά, οἱ δ’ (οἱ Πυθαγόρειοι) ἀριθμοὺς εἶναί φασιν αὐτὰ τὰ αἰσθητά. [Sidenote: The elements of numbers.] 144. Aristotle speaks of certain “elements” (στοιχεῖα) of numbers, which were also the elements of things. That, of course, is only his own way of putting the matter; but it is clearly the key to the problem, if we can discover what it means. Primarily, the “elements of number” are the Odd and the Even, but that does not seem to help us much. We find, however, that the Odd and Even were identified in a somewhat violent way with the Limit and the Unlimited, which we have seen reason to regard as the original principles of the Pythagorean cosmology. Aristotle tells us that it is the Even which gives things their unlimited character when it is contained in them and limited by the Odd,[783] and the commentators are at one in understanding this to mean that the Even is in some way the cause of infinite divisibility. They get into great difficulties, however, when they try to show how this can be. Simplicius has preserved an explanation, in all probability Alexander’s, to the effect that they called the even number unlimited “because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on _ad infinitum_. But, when the odd is added, it limits it; for it prevents its division into equal parts.”[784] Now it is plain that we must not impute to the Pythagoreans the view that even numbers can be halved indefinitely. They had carefully studied the properties of the decad, and they must have known that the even numbers 6 and 10 do not admit of this. The explanation is really to be found in a fragment of Aristoxenos, where we read that “even numbers are those which are divided into equal parts, while odd numbers are divided into unequal parts and have a middle term.”[785] This is still further elucidated by a passage which is quoted in Stobaios and ultimately goes back to Poseidonios. It runs: “When the odd is divided into two equal parts, a unit is left over in the middle; but when the even is so divided, an empty field is left, without a master and without a number, showing that it is defective and incomplete.”[786] Again, Plutarch says: “In the division of numbers, the even, when parted in any direction, leaves as it were within itself ... a field; but, when the same thing is done to the odd, there is always a middle left over from the division.”[787] It